Geometrical derivation of Lagrange’s equations for a system of particles

Abstract: A geometrical derivation is given for Lagrange’s equations for a system of rigid bodies subject to general holonomic and non-holonomic constraints. As in the case of a similar derivation for a system of par-ticles, the entire system is represented by an abstract particle P moving in a higher-dimensional Euclidean space, called Hertzian space, the metric of which is determined by the radius of gyration of the physical sys-tem. The holonomic constraints confine P to move in a Riemannian manifold – the configuration manifold of the constrained system – embedded in Hertzian space. Euler’s laws of linear and angular momenta are expressed as a single balance equation in Hertzian space and Lagrange’s equations emerge as covariant com-ponents of this equation taken along the coordinate directions in the configuration manifold. No appeal is made to variational principles or to notions of virtual work. Key Words: Lagrange’s equations, system of rigid bodies 1